Methuselah

A methuselah is a pattern that takes a large number of generations in order to stabilize (known as its lifespan) and becomes much larger than its initial configuration at some point during its evolution. There is no consensus on the exact definition,[1] but patterns that stabilize in less than 100 generations are not generally called methuselahs.

Martin Gardner defined methuselahs as patterns of fewer than ten cells that take longer than 50 generations to stabilize.[2] Some other interpretations allow for more cells while requiring a longer lifespan, or characterize the size of an initial configuration by the size of its bounding box instead of the number of cells. Others use more complex metrics to measure the "quality" of methuselahs (see Measuring methuselahs below).

The time when a pattern is considered to have stabilized is commonly agreed upon to be the first generation such that the pattern can be resolved into still lifes, oscillators and escaping spaceships, provided such a generation exists. For infinitely growing patterns, no agreed-upon definition is known, although the Life Lexicon describes a particular ark as stabilizing at generation 736692.[3] Most interpretations exclude such patterns.

There is no limit to the lifespan of a pattern with 8 or more cells, as a methuselah consisting of a glider heading towards an arbitrarily distant blinker can be trivially constructed. Therefore, patterns with excessively large bounding boxes are generally implicitly excluded.

Contents

Examples

The smallest and most well-known methuselah is the R-pentomino, a pattern of five cells first considered by John Conway[4] that takes 1103 generations before stabilizing as a pattern of eight blocks, six gliders, four beehives, four blinkers, one boat, one loaf, and one ship. This methuselah is particularly notable since almost all other patterns of similar size stabilize within 10 generations.

Versions v4.54 and above of apgsearch report soups lasting at least 25,000 generations, allowing the results to be tabulated on Catagolue.[7] As of early 2019, the longest-lasting methuselah found using apgsearch takes 47575 generations to stabilize and was found by Adam P. Goucher on February 10, 2019.[8] Versions v4.69 and above also report diehards lasting at least 500 generations, referring to them as "messless methuselahs".[9][note 1] After v5.03, apgsearch also reports soups with a stabilization population of above 3000 in a category of "megasized methuselae".

Due to the difficulty of testing a soup's ash for stability, both of these censuses estimate the lifespan of methuselahs found.[note 2][note 3]

Measuring methuselahs

Various metrics have been proposed to measure methuselahs so as to reward patterns such as the R-pentomino and acorn while penalizing trivial examples such as the glider-and-blinker construction mentioned above. Oscar Cunningham suggested using the minimum covering polyplet size (MCPS) for this as a compromise between population and bounding box,[10] resulting in the L/MCPS metric, the quotient of the methuselah's lifespan and its MCPS.

Other quotient-based metrics include F/I, F/L, and L/I, with F, I, and L standing for final population, initial population, and lifespan respectively.

See also

Notes

↑Long-lived soups found as part of TOLLCASS had their exact lifespan verified manually.

↑apgsearch automatically tests the lifespan of a soup more precisely if its estimated lifespan is sufficiently high, but is not guaranteed to detect all methuselahs with a lifespan of less than 26,000 generations.